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Coincidence of algebraic and smooth theta correspondences


Authors: YiXin Bao and BinYong Sun
Journal: Represent. Theory 21 (2017), 458-466
MSC (2010): Primary 22E46, 22E50
DOI: https://doi.org/10.1090/ert/508
Published electronically: November 1, 2017
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Abstract: An ``automatic continuity'' question has naturally occurred since Roger Howe established the local theta correspondence over $ \mathbb{R}$: Does the algebraic version of local theta correspondence over $ \mathbb{R}$ agree with the smooth version? We show that the answer is yes, at least when the concerning dual pair has no quaternionic type I irreducible factor.


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Additional Information

YiXin Bao
Affiliation: School of Sciences, Harbin Institute of Technology, Shenzhen, 518055, China
Email: baoyixin@hit.edu.cn

BinYong Sun
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences –and– School of Mathematics, University of Chinese Academy of Sciences, Beijing, 100190, China
Email: sun@math.ac.cn

DOI: https://doi.org/10.1090/ert/508
Keywords: Reductive dual pair, theta correspondence, Casselman-Wallach representation, Harish-Chandra module.
Received by editor(s): January 13, 2017
Received by editor(s) in revised form: May 26, 2017, and August 10, 2017
Published electronically: November 1, 2017
Additional Notes: The second author was supported in part by the National Natural Science Foundation of China (No. 11525105, 11688101, 11621061 and 11531008).
Article copyright: © Copyright 2017 American Mathematical Society

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